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Question:

Suppose that the linear system $$\begin{bmatrix} 2 & 0 & -1 \\ 0 & c & 1 \\ 1 & 3 & -2 \end{bmatrix} \vec{x}=\vec{b}= \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$$ is inconsistent for some vector $\vec{b}$. Determine c.

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What I did:

I put this into RREF and got :

$$\left[\begin{array}{rrr|r} 2 & 0 & -1 & x_1\\ 0 & c & 1 & x_2 \\ 0 & 2+c & 0 & 2x_3-x_1+x_2 \end{array}\right]$$

I do not know what to do next.

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  • $\begingroup$ When do you get a row of all zero coefficients? (If you already know determinants, that is another option.) $\endgroup$ – Daniel Fischer Jun 20 '15 at 21:58
  • $\begingroup$ @DanielFischer c=-2 ? :D $\endgroup$ – The Artist Jun 20 '15 at 21:59
  • $\begingroup$ @DanielFischer but $b$ could be the zero vector right? No restriction has been mentioned in the question $\endgroup$ – The Artist Jun 20 '15 at 22:02
  • $\begingroup$ The question is when the system "is inconsistent for some vector $b$". Every linear system with right hand side $0$ is consistent. Also every system without a zero coefficient row in the RREF. The question is when you can find some $b$ such that the system is inconsistent. $\endgroup$ – Daniel Fischer Jun 20 '15 at 22:05
  • $\begingroup$ @DanielFischer thank you :) $\endgroup$ – The Artist Jun 20 '15 at 22:11
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The system is inconsistent for some vector $b$ if and only if its rank is $<3$, since any system of rank $3$ is consistent, i. e. the linear map associated with the matrix of the system is an isomorphism.

Let's row reduce the matrix of the system: $$\begin{bmatrix}2&0&-1\\0&c&1\\1&3&-2\end{bmatrix}\rightsquigarrow\begin{bmatrix}2&0&-1\\0&c&1\\0&6&-3\end{bmatrix}\rightsquigarrow\begin{bmatrix}2&0&-1\\0&6&-3\\0&c&1\end{bmatrix}\rightsquigarrow\begin{bmatrix}2&0&-1\\0&6&-3\\0&0&3c+6\end{bmatrix} $$ We see the system has rank $<3$ (and indeed has rank $2$) if and only if $3c+6=0$, i. e. $\,c=-2$.

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